Particle Weights and their Disintegration II

TL;DR

This paper develops a disintegration theory for particle weights, linking them to irreducible representations and improper energy-momentum eigenstates, using separable constructs to handle non-separable GNS-representations.

Contribution

It introduces a disintegration approach for particle weights that connects them to pure components and irreducible representations, overcoming non-separability issues.

Findings

Disintegration yields pure components linked to irreducible representations.

The approach handles non-separable GNS-representations via restricted separable constructs.

Representations are locally normal and extendable under reasonable phase space assumptions.

Abstract

The first article in this series presented a thorough discussion of particle weights and their characteristic properties. In this part a disintegration theory for particle weights is developed which yields pure components linked to irreducible representations and exhibiting features of improper energy-momentum eigenstates. This spatial disintegration relies on the separability of the Hilbert space as well as of the C*-algebra. Neither is present in the GNS-representation of a generic particle weight so that we use a restricted version of this concept on the basis of separable constructs. This procedure does not entail any loss of essential information insofar as under physically reasonable assumptions on the structure of phase space the resulting representations of the separable algebra are locally normal and can thus be continuously extended to the original quasi-local C*-algebra.